Ordering positive and negative numbers using inequality symbols

Tuesday morning, the weather app says minus two degrees. Wednesday morning, same week: minus nine. One of those mornings you can get away with a jumper. The other one, you really cannot. Nine is bigger than two. You have known that since you were small. But minus nine is the colder morning, and colder means lower. So the number with the bigger digit turned out to be the smaller number. That flip is what this topic is about.

Everything here hangs off one picture. Let's put it on screen and leave it there.

A number line. Zero sits in the middle. The positive numbers run off to the right: one, two, three, and on. The negative numbers run off to the left: minus one, minus two, minus three. Every step is the same size. Those whole numbers, going both ways, have a name. Integers. An integer is a whole number, and it can be positive, negative, or zero. Zero is the odd one out: not positive, not negative. It is the join between the two sides. One word thing while we are here. The little dash in front of a number is read out loud as minus. Minus nine. As a group, the numbers living left of zero are called negative numbers. Now the rule that never breaks. On this line, further right is bigger, and further left is smaller. Always. It makes no difference what the digits look like. Minus nine sits nine steps below zero. Minus two sits only two steps below. So minus nine is further left, and further left is less.

Quick one, and it is a gentle one. Three numbers: minus one, minus eight, and zero. Which of them sits furthest left on the line? A: minus one. B: minus eight. C: zero. Which one is furthest left? Have a think. It is B, minus eight. It sits eight steps below zero, so it lands further left than minus one, which is only one step below. Furthest left also means smallest, so minus eight is the smallest of the three.

Hold on to that, because it carries through this whole chapter. Where it sits is what it means. A number's position on the line tells you its size. How big its digits look does not.

That is the ordering. Now the symbols you use to write a comparison down.

Three to start with. The equals sign says two things are the same value. Then two pointed ones: less than, and greater than. Each has an open end and a sharp end. Here is how to read them, whichever way round they turn up. Read the statement left to right, like a sentence. The open end always faces the bigger number. The sharp point always faces the smaller one. The symbol is a picture of the gap: wide at the big end, pinched at the small end. Five, symbol, two, open end against the five. Left to right, that says five is greater than two. Now the same fact the other way round. Two, symbol, five, open end still against the five. That says two is less than five. Same symbol, turned around. Two more, the same shapes with a flat line underneath. That line means or equal to. One reads is less than or equal to, the other is greater than or equal to. Picture a lift with a sign saying up to eight people. The number of people is less than or equal to eight. Eight is allowed. Nine is not.

Time to do a real one. Five numbers, all mixed up.

Order these five numbers, starting with the smallest. Four. Minus nine. Zero. Minus two. Seven. Pause the video here and have a go at it. I'll wait. The answer is minus nine, minus two, zero, four, seven. Here is how to get there without guessing. Mark each number on the line. Minus nine a long way left, minus two a short way left, zero in the middle, four and seven to the right. Then you do not order anything. You read the line left to right and write down what you pass. Two things about writing that answer out. The question said starting with the smallest, so that is the direction you write in. Had it said largest first, same list, backwards. And write the numbers themselves, exactly as handed to you, not their positions in the list.

Same idea, now with the symbols doing the talking.

Three statements to complete, each with less than, greater than, or equals. Part a: minus three, gap, minus seven. Part b: minus nought point six, gap, minus nought point eight. Part c: two fifths, gap, nought point three. Pause the video and fill in all three of them. I'll wait. Part a is minus three is greater than minus seven. Check it on the line. Minus seven sits seven steps below zero, minus three only three below, so minus three is further right, and further right is bigger. The seven looks bigger, and that look is the whole trap. Part b runs the same way, and decimals change nothing. Nought point eight is further from zero than nought point six, so minus nought point eight sits further left. Minus nought point six is greater than minus nought point eight. Part c has no negatives at all. Two fifths means five equal parts with two taken. Half of five parts would be two and a half, so two parts sits just under half. Nought point three is three tenths, and half is nought point five, so that sits well under half. Both are under half, and two fifths is the closer one, so two fifths is greater than nought point three. One habit for these. Once the symbol is in, read the line back left to right as a sentence and ask if it is true. Minus three is greater than minus seven. True. That catches a symbol pointing the wrong way. And leave each number in the form you were given it. Two fifths stays a fraction.

Let's look at what examiners have written about this exact kind of question. An examiner report says this. Most students were able to order a list of five integers in part a and five decimals in part b. Where a mark was not scored with the integers, it was by putting the negative numbers in the wrong order. Read that carefully, because most of it is good news. The ordering itself was fine. The direction, the positive numbers, fine. One spot did the damage, and it was the negatives. So the trap has a shape you can watch for. It turns up whenever two negative numbers land in one list, because that is the one place where bigger-looking digits belong to the smaller number. So before you leave any ordering answer, do one pass over the negatives alone. Are they in the order you would meet them walking left to right along the line?

Right, your go. This one is on your own this time.

Put these in order, smallest first, and no calculator. Five. Minus six. Minus one. Three. Zero. Pause the video and write your list out. I'll wait. The answer is minus six, minus one, zero, three, five. Mark them on the line, read left to right, write what you pass. Then check the two negatives against each other. Minus six is six steps below zero, minus one only one step below, so minus six is further left, and it goes first.

Gathering it up now. Four things, and every one of them comes off the same picture. One. An integer is a whole number: positive, negative or zero. Two. On the number line, further right is bigger and further left is smaller, always. Three. Read a comparison left to right like a sentence: the open end faces the bigger number, the sharp point the smaller. Four. Two negative numbers in one list is where your care goes, because that is where bigger-looking digits sit on the smaller number. And underneath all four sits the line for this chapter. Where it sits is what it means. Minus nine is not smaller than minus two because of anything about nine and two. It is smaller because of where it sits. Further left. Further below zero. The colder morning.

Next in the chapter: Place value and decimal notation. You can now say which of two numbers is the bigger one. That video takes a single number apart and shows you where each digit's value comes from.

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